The present invention relates to fiber optic gyroscopes used for rotation sensing and, more particularly, to resonator fiber optic gyroscopes.
Fiber optic gyroscopes are an attractive means with which to sense rotation. They can be made quite small and still be constructed to withstand considerable mechanical shock, temperature change, and other environmental extremes. In the absence of moving parts, they can be nearly maintenance free, and they have the potential to become economical in cost. They can also be sensitive to low rotation rates that can be a problem in other kinds of optical gyroscopes.
There are various forms of optical inertial rotation sensors which use the well known Sagnac effect to detect rotation about a pertinent axis thereof. These include active optical gyroscopes having the gain medium contained in an optical cavity therein, such as the ring laser gyroscope, and passive optical gyroscopes without any gain medium in the primary optical path, such as the interferometric fiber optic gyroscope and the ring resonator fiber optic gyroscope. The avoidance of having the active medium along the primary optical path in the gyroscope eliminates some problems which are encountered in active gyroscopes such as low rotation rate lock-in, bias drift and some causes of scale factor variation.
Interferometric fiber optic gyroscopes typically employ a single spatial mode optical fiber of a substantial length formed into a coil, this substantial length of optical fiber being relatively costly. Resonator fiber optic gyroscopes, on the other hand, are constructed with relatively few turns of a single spatial mode optical fiber giving them the potential of being more economical than interferometric fiber optic gyroscopes. A resonator fiber optic gyroscope typically has three to fifty meters of optical fiber in its coil versus 100 to 2,000 meters of optical fiber in coils used in interferometric fiber optic gyroscopes. In addition, resonator fiber optic gyroscopes appear to have certain advantages in scale factor linearity and dynamic range.
In either type of passive gyroscope, these coils are part of a substantially closed optical path in which an electromagnetic wave, or light wave, is introduced and split into a pair of such waves, to propagate in opposite directions through the optical fiber coil to both ultimately impinge on a photodetector or photodetectors, a single photodetector for both waves in interferometric fiber optic gyroscopes and on corresponding ones of a pair of photodetectors in resonator fiber optic gyroscopes. Rotation about the sensing axis of the core of the coiled optical fiber in either direction provides an effective optical path length increase in one rotational direction and an effective optical path length decrease in the opposite rotational direction for one member of this pair of electromagnetic waves. The opposite result occurs for the remaining member of the pair of electromagnetic waves for such rotation. Such path length differences between the pair of electromagnetic waves introduce corresponding phase shifts between those waves in interferometric fiber optic gyroscopes, or corresponding different optical cavity effective optical path lengths for these waves in a resonator fiber optic gyroscope.
In this latter instance, one or more optical frequency shifters are used to each effectively adjust the frequency of a corresponding one of the pair of electromagnetic waves that circulate in opposite directions in the resonator fiber optic coil. This is accomplished through such a frequency shifter shifting the frequency of a corresponding input electromagnetic wave giving rise to the resonator electromagnetic wave of interest. As a result, through feedback arrangements, the frequencies of each member of the pair of electromagnetic waves can be kept in resonance with the effective optical path length that wave is experiencing in the resonator fiber optic coil. Hence, any frequency difference between these waves becomes a measure of the rotation rate experienced by the resonator fiber optic coil about the axis around which this coil has been positioned. In such resonances, each wave has the portions thereof that previously were introduced in the resonator coil and have not yet dissipated, and the portions thereof currently being introduced in the resonator coil, at a frequency such that they are all in phase with one another so they additively combine to provide a peak in the intensity of that wave in that resonator over a local range of frequencies.
The difference in frequency between the members of the pair of opposing electromagnetic waves in a resonant fiber optic gyroscope is desired to be constant when rotation conditions about the resonator optic fiber coil axis are unchanging thereby requiring that stable resonance conditions occur in that resonator in those circumstances. Furthermore, there are several advantages in achieving frequency shifting of the resonator electromagnetic waves by operating one or more integrated optics phase modulators for this purpose through each of which the corresponding input electromagnetic wave is transmitted. These advantages involve economics, packaging volume, and performance. Obtaining a constant frequency difference between these resonator wave pair members using such a phase modulator requires that the phase modulator change phase in the form of a linear ramp since the derivative of phase with respect to time yields the frequency.
Because of the impossibility of having a phase modulator provide an infinite duration linear ramp with respect to time, a repetitive linear ramp with periodic resetting of the phase to a reference value must be used. The resulting sawtooth phase change waveform results in what is termed serrodyne phase modulation of those electromagnetic waves passing through the modulator.
If the resetting of the serrodyne waveform from the ramp peak value, reached at the end of each linear ramp, back to an initial value is instantaneous, then, of course, the succeeding ramp begins immediately at the end of the preceding ramp. As a result, effectively, there is no change in frequency value based on the time derivative of the succeeding linear ramp from the value obtained from the time derivative of the preceding ramp. In practice, the reset time is not zero, and some high frequency signal components are introduced by the high rate of change of phase during such a reset. This results in unwanted harmonics, any unwanted effects of which must be avoided in the design of the system.
A further practical matter in connection with the serrodyne waveform is the magnitude of the phase change which occurs during such a reset. A phase change in one of the pair of input electromagnetic waves that equals a shift of one period of the frequency of the wave, or 2.pi., leaves that input electromagnetic wave effectively in phase with the corresponding resonator electromagnetic wave. Hence, coupling that reset, or shifted, input wave into the resonator coil causes no changes to occur in the corresponding electromagnetic wave in resonance there. However, a shift of a different magnitude which may be close to, but not equal to, 2.pi. means the portions of the input electromagnetic wave coupled into the resonator coil will not be in phase with the electromagnetic wave previously resonating there. This situation will lead to a relatively gradual changing of the phase of that resonating wave to the new value resulting from the reset as more of the input wave is coupled into the resonator coil. This resonator wave phase changing will continue until the input wave and the resonator wave are again effectively in phase, with the duration of the out of phase condition initially resulting from the reset leading to a similar duration of an error in the corresponding photodetector output signal.
Consider the known resonator fiber optical gyroscope system of FIG. 1. An optical cavity resonator, 10, formed by a continual path optical fiber is provided with an input directional coupler, 11, and an output directional optical coupler, 12. Resonator 10 is formed of a single spatial mode optical fiber which has two polarization eigenstates. Avoiding different optical path lengths for electromagnetic waves in each state is solved by thoroughly mixing the polarized waves in each state. In the first instance, such mixing is achieved by fabricating the resonator coil with two ends of a three to fifty meter length of such fiber spliced together so that the birefringence principal axes of the fiber are rotated 90.degree. with respect to each other on opposite sides of the splice. The resonator fiber is characterized by a loss coefficient, .alpha., and an average of the propagation constants for the principal birefringence axes, .beta..sub.o, assuming an ideal 90.degree. splice.
Directional coupler 11 is fabricated by appropriately fusing together an input optical fiber, 13, with the optical fiber in resonator 10 with the fibers being tapered as they come into the fused portion on either side of that portion. Directional coupler 11 provides a phase shift of .pi./2 between an input electromagnetic wave and the resulting electromagnetic wave at the resonator output thereof, the output wave further being characterized with respect to the input electromagnetic wave by a coupler coupling coefficient, k.sub.1, and a coupler loss coefficient, .gamma..sub.1. Directional coupler 11 has a suitable packaging arrangement thereabout.
Directional coupler 12 is constructed in generally the same manner as is directional coupler 11, but here an output optical fiber, 14, is fused to the optical fiber of resonator 10. Directional coupler 12 is characterized by a coupler coupling coefficient, k.sub.2, and a coupler loss coefficient, .gamma..sub.2.
The opposite ends of input optical fiber 13 are each connected to an integrated optics chip, 15, formed with lithium niobate (LiNbO.sub.3) as the base material therefor. These ends of fiber 13 are appropriately coupled to integrated optical waveguides, 16 and 17, formed in the base material of integrated optics chip 15. The relationship of the ends of input optical fiber 13 and the ends of integrated waveguides 16 and 17 are such that electromagnetic waves can be efficiently passed therebetween without undue losses. Integrated waveguide 16 is provided between a pair of metal plates formed on the base material of integrated optics chip 15 to provide a phase modulator, 18, therein. Similarly, integrated waveguide 17 is formed between a another pair of metal plates formed on the base material to result in a further phase modulator, 19, in integrated optics chip 15. Integrated waveguides 16 and 17 merge with one another into a single integrated waveguide, 20, to thereby provide a "Y" coupler in integrated optics chip 15.
A laser, 21, is coupled to integrated waveguide 20 in a suitable manner so that light may be transmitted efficiently from laser 21 to integrated waveguide 20. Laser 21 is typically a solid state laser emitting electromagnetic radiation having a wavelength of 1.3 .mu.m with a spectral line width of one to hundreds of Khz. The wavelength at which laser 21 operates, or the frequency thereof, f.sub.o, can be adjusted by signals at an input thereof. Typical ways of providing such adjustment is to control the temperature of, or the current through, the solid state laser, or through the "pumping" semiconductor light emitting diode for the solid state laser, which in the latter instance may be a Nd:Yag laser. Where the diode is the emitting laser, the laser type may be an external cavity laser, a distributed feedback laser or other suitable types.
Thus, electromagnetic radiation emitted by laser 21 at a variable frequency f.sub.o is coupled to integrated waveguide 20, and from there split into two portions to form a pair of electromagnetic waves traveling in the input optical path in directions opposite one another. That is, the electromagnetic wave portion transmitted through integrated waveguide 16 proceeds therethrough and past phase modulator 18 into input optical fiber 13, and through input directional coupler 11 where a fraction k.sub.1 is continually coupled into resonator 10 to repeatedly travel therearound in a first direction, the counterclockwise direction, there being a continual fractional loss for that wave of .gamma..sub.1 in coupler 11 as indicated above. The remaining portion of that wave, neither entering resonator 10 nor lost in coupler 11, continues to travel along input optical fiber 13 into integrated optical waveguide 17, through phase modulator 19, and finally through integrated waveguide 20 returning toward laser 21 (usually stopped by an isolator).
Similarly, the electromagnetic wave portion from laser 21, entering integrated waveguide 20 to begin in integrated waveguide 17, passes through phase modulator 19 into input optical fiber 13 and input directional coupler 11 where a fraction k.sub.1 thereof is continually coupled into resonator 10, accompanied by a continual fractional loss of .gamma..sub.1, to repeatedly traverse resonator 10 in a direction opposite (clockwise) to that traversed by the first portion coupled into resonator 10 described above. The remaining portion not coupled into resonator 10, and not lost in directional coupler 11, continues through input optical fiber 13 into integrated waveguide 16, passing through phase modulator 18, to again travel in integrated waveguide 20 in the opposite direction on its return toward laser 21.
The pair of opposite direction traveling electromagnetic waves in resonator 10, a clockwise wave and a counterclockwise wave, each have a fraction k.sub.2 continually coupled into output optical fiber 14 with a fraction .gamma..sub.2 of each continually lost in coupler 12. The counterclockwise wave is transmitted by coupler 12 and fiber 14 to a corresponding photodetector, 22, and the clockwise wave is transmitted by them to a corresponding photodetector, 23, these photodetectors being positioned at opposite ends of output optical fiber 14. Photodetectors 22 and 23 are typically p-i-n photodiodes each of which is connected in corresponding one of a pair of bias and amplifying circuits, 24 and 25, respectively.
The frequency of the electromagnetic radiation emitted by laser 21, after being split from its combined form in integrated waveguide 20 into separate portions in integrated waveguides 16 and 17, has each of its portions shifted from frequency f.sub.o to a corresponding resonance frequency by phase modulators 18 and 19, respectively. The portion of the electromagnetic wave diverted into integrated waveguide 16 is shifted from frequency f.sub.o to frequency f.sub.o +f.sub.1 by phase modulator 18, and this frequency shifted electromagnetic wave is then coupled by input directional coupler 11 into resonator 10 as the counterclockwise electromagnetic wave. Similarly, the portion of the electromagnetic wave directed into integrated waveguide 17 from integrated waveguide 20 is shifted in frequency from f.sub.o to f.sub.o +f.sub.2 by phase modulator 19 from whence it is coupled into resonator 10 by input directional coupler 11 to form the clockwise wave therein. These shiftings of frequency are caused by serrodyne waveforms applied to phase modulators 18 and 19, the serrodyne waveform for phase modulator 18 being supplied from a controlled serrodyne generator, 26, and the serrodyne waveform for phase modulator 19 being provided by a serrodyne generator, 27.
Thus, controlled serrodyne generator 26 provides a sawtooth waveform output signal having a repetitive linear ramp variable frequency f.sub.1, the frequency f.sub.1 of this sawtooth waveform being controlled by an input shown on the left side of generator 26 in FIG. 1. The repetitive linear ramp frequency of the sawtooth waveform from serrodyne generator 27 is fixed, and is held at a constant value, f.sub.2.
The clockwise electromagnetic wave in resonator 10 and the counterclockwise electromagnetic wave in resonator 10 must always have the frequencies thereof driven toward values causing these waves to be in resonance in resonator 10 for the effective optical path length each is experiencing. This includes the path length variation resulting from any rotation of resonator 10 about the symmetrical axis thereof that is substantially perpendicular to the plane of the loop forming that optical resonator. Since controlled serrodyne generator 26 has the frequency of its serrodyne waveform controlled externally, that frequency value can be adjusted to the point that the corresponding counterclockwise wave in resonator 10 is in resonance with its effective path length, at least in a steady state situation. There, of course, can be transient effects not reflecting resonance in situations of sufficiently rapid changes of rotation rates of resonator 10. On the other hand, the constant frequency of the sawtooth waveform of serrodyne generator 27 requires that the clockwise electromagnetic wave in resonator 10 be adjusted by other means, the means chosen here being adjustment of the frequency value of the light in laser 21. Thus, the adjustment of the value of the frequency f.sub.1 of the sawtooth waveform of controlled serrodyne generator 26 can be accomplished independently of the adjustment of the frequency f.sub.o of laser 21 so that, in steady state situations, both the counterclockwise electromagnetic wave and the clockwise electromagnetic wave in resonator 10 can be in resonance therein despite each experiencing a different effective optical path length therein.
Adjusting the frequency of the counterclockwise and clockwise electromagnetic waves traveling in opposite directions in resonator 10 means adjusting the frequency of each of these waves so that they are operating at the center of one of the peaks in the corresponding intensity spectra for resonator 10 experienced by such waves. Maintaining the frequency of the counterclockwise and the clockwise waves at the center of a corresponding resonance peak in the corresponding one of the resonator intensity spectra would be a difficult matter if that peak had to be estimated directly without providing some additional indicator of just where the center of the resonance peak actually is. Thus, the system of FIG. 1 introduces a bias modulation with respect to each of the counterclockwise and clockwise waves in resonator 10 through phase modulators 18 and 19, respectively. Such a bias modulation of each of these waves is used in a corresponding feedback loop to provide a loop discriminant characteristic followed by a signal therein which is acted on by that loop to adjust frequency f.sub.o and f.sub.1 as necessary to maintain resonance of the clockwise and counterclockwise waves, respectively.
A bias modulation generator, 28, provides a sinusoidal signal at a frequency f.sub.m which is added to the sawtooth waveform of frequency f.sub.1 provided by controlled serrodyne generator 26 and that is used to shift the frequency of the counterclockwise wave in resonator 10. Similarly, a further bias modulation generator, 29, provides a sinusoidal waveform of a frequency f.sub.n which is added to the sawtooth waveform at frequency f.sub.2 provided by serrodyne generator 27. Frequencies f.sub.m and f.sub.n differ from one another to reduce the effects of electromagnetic wave backscattering in the optical fiber of resonator 10. The adding of the sinusoidal signal provided by bias modulation generator 28 and the sawtooth waveform of controlled serrodyne generator 26 is accomplished in an appropriate summer, 30. The addition of the sinusoidal signal provided by bias modulator generator 29 to the sawtooth waveform provided by serrodyne generator 27 is accomplished in a further summer, 31.
The combined waveform provided at the output of summer 30 is amplified in a power amplifier, 32, which is used to provide sufficient voltage to operate phase modulator 18. Similarly, the combined output signal provided by summer 31 is provided to the input of a further power amplifier, 33, used to provide sufficient voltage to operate phase modulator 19. Thus, the input electromagnetic wave to resonator 10 from integrated waveguide 16 will contain instantaneous electric field component frequencies of: EQU f.sub.o +f.sub.1 -f.sub.m .DELTA..phi..sub.m sin .omega..sub.m t
where .DELTA..phi..sub.n is the amplitude of the bias modulation phase change at frequency f.sub.n.
The fraction of the electromagnetic wave reaching photodetector 22 through resonator 10 is not only shifted in frequency to a value of f.sub.o +f.sub.1, but is also frequency modulated at f.sub.m. Depending on the difference between the resonance frequency and f.sub.o +f.sub.1, the intensity at that photodetector will thus have variations occurring therein at integer multiples of f.sub.m (though none will occur at exact resonance). These latter components have amplitude factors related to the deviation occurring in the sum of (a) the phase shift resulting from the propagation constant multiplied by the path length in the counterclockwise direction in resonator 10, plus (b) phase shifts due to rotation and other sources, from a value equaling an integer multiple of 2.pi., a condition necessary for resonance along the effective optical path length in this direction.
Similarly, field components of electromagnetic waves in integrated waveguide 17 enroute to resonator 10 will have instantaneous frequency components equal to: EQU f.sub.o +f.sub.2 -f.sub.n .DELTA..phi..sub.n sin.omega..sub.n t
where .DELTA..phi..sub.n is the amplitude of the bias modulation phase change at frequency f.sub.n.
The fraction thereof reaching photodetector 23 through resonator 10 is again shifted in frequency to a value in this instance of f.sub.o +f.sub.2 and frequency modulated at f.sub.n. Again, the intensity at photodetector 23 will have variations therein at integer multiples of f.sub.n if these clockwise waves are not at exact resonance. These latter components also have amplitude factors related to the deviation of the sum of (a) the phase shift resulting from the propagation constant multiplied by the path length in the clockwise direction in resonator 10, plus (b) phase shifts due to rotation and other sources, from a value equaling an integer multiple of 2.pi., again, a condition necessary for resonance along the effective optical path length in that direction.
Since the output signal of photodetector 22 has a frequency component at f.sub.m that is a measure of the deviation from resonance in resonator 10 in the counterclockwise direction, the output signal of bias and amplifier photodetector circuit 24 is provided to a filter, 34, capable of passing signal portions having a frequency component f.sub.m. Similarly, the output signal of photodetector 23 has a frequency component at f.sub.n that is a measure of the deviation from resonance in the clockwise direction, and so a filter, 35, is provided at the output of photodetector bias and amplifier circuit 25 capable of passing signal components having a frequency of f.sub.n.
The output signal from filter 34 is then provided to a phase detector, 36, at an operating signal input thereof. Phase detector 36 is a phase sensitive detector which also receives, at a demodulation signal input thereof, the output signal of bias modulation generator 28 which is the sinusoidal signal at frequency f.sub.m. Similarly, the output signal from filter 35 is provided to an operating signal input of a further phase detector, 37, which also receives at a demodulation input thereof the output sinusoidal signal at frequency f.sub.n of bias modulation generator 29. The output signals of phase detectors 36 and 37 follow a loop discriminant characteristic so that they indicate how far from resonance are the corresponding frequencies in resonator 10.
The discriminant characteristic followed by the output of phase detectors 36 and 37 will change algebraic sign for the frequencies on either side of the resonance peak and will have a zero magnitude at the resonance peak or resonance center. In fact, for sufficiently small values of the bias modulation generator output signals, the characteristic followed by the output signals of phase detectors 36 and 37 will be close to the derivative with respect to frequency of the intensity spectrum near the corresponding resonance peak. Thus, the output characteristics followed by the output signals of phase detectors 36 and 37 provide signals well suited for a feedback loop used to adjust frequencies to keep the corresponding electromagnetic waves in resonance in resonator 10.
Errors in the feedback loop are to be eliminated, and so the output signal of phase detector 36 is supplied to an integrator, 38, and the output signal of phase detector 37 is supplied to a further integrator, 39. Deviations from resonance are stored in these integrators which are then used in the loop to force the waves back to resonance in resonator 10. The output signal of integrator 38, in turn, is supplied to an amplifier, 40, used to provide signals to the modulation input of controlled serrodyne generator 26, thus completing the feedback loop for adjusting serrodyne frequency f.sub.1. Similarly, the output signal of integrator 39 is supplied to an amplifier, 41, which in turn has its outputs supplied to laser 21 to control the frequency f.sub.o of light being emitted by laser 21, thereby closing the remaining feedback loop.
This result can be better seen by considering in more detail the effects on one member of the input electromagnetic wave pair provided by laser 21 in input optical fiber 13 through its transmitting electromagnetic waves to integrated waveguide 20 that are subsequently diverted into two portions, one each in integrated waveguides 16 and 17 to each thus become the basis of one member of this pair. Consider the electromagnetic field component of the electromagnetic wave portion that has passed into integrated waveguide 16, where it becomes subject to the influence of phase modulator 18 as it passes to input fiber 13 and then to input directional coupler 11 to enter, in part, resonator 10. Assume that the electric field component of the electromagnetic wave from laser 21 can be written as E.sub.in e.sup.i.omega.ot, and that a fraction q thereof is transmitted into integrated waveguide 16. As a result of the phase modulation in phase modulator 18, there will be a shift f.sub.1 in the frequency of the electromagnetic wave portion represented by electric field component qE.sub.i due to controlled serrodyne generator 26, and a phase change therein, .DELTA..phi. cos.omega..sub.m t, due to bias modulation generator 28 supplying a sinusoidal signal reaching phase modulator 18. Thus, the electromagnetic wave portion electric field component entering input directional coupler 11 may be represented as qE.sub.in e.sup.i(.omega..sbsp.o.sup.+.omega..sbsp.1.sup.)t e.sup.i.DELTA..phi.cos.omega..sbsp.m.sup.t where .omega..sub.o +.omega..sub.1 =2.pi.(f.sub.o +f.sub.1). The static phase shift due to the propagation of light between laser 21 and coupler 11 has been neglected for simplicity since no significant effect would follow therefrom below.
A fraction .sqroot.k.sub.1 of this electromagnetic wave portion field component enters resonator 10 through input directional coupler 11 with a 90.degree. phase shift, and of this only a fraction .sqroot.1-.gamma..sub.1 reaches resonator loop 10. Although electromagnetic waves with polarizations other than the eigenstate polarization of resonator 10 will enter coupler 11 and resonator 10, they will not be in resonance in resonator 10 and so will represent relatively little energy therein. The part of the input electromagnetic wave portion electric field component entering resonator 10 travels a distance l.sub.1 therein to reach output directional coupler 12.
A fraction .sqroot.k.sub.2 (1-.gamma..sub.2) of that electromagnetic wave field component reaching output directional coupler 12 is then coupled through another 90.degree. phase shift to output optical fiber 14. The part remaining in resonator 10 then travels a distance l.sub.2, over which it passes through the resonator splice indicated above with a phase shift .theta., to again reach input directional coupler 11 where a further fraction is lost therein and another fraction is coupled out of input directional coupler 11 to input optical fiber 13, with the remainder again traveling to output coupler 12, etc.
The foregoing process in resonator 10, and couplers 11 and 12, and input and output fibers 13 and 14 leads to an electromagnetic wave electric field component E.sub.d reaching photodetector 22 based on the electromagnetic wave portion in integrated waveguide 16. The electromagnetic wave electric field component E.sub.d in the polarization state resonant in resonator 10 reaching photodetector 22, after traveling along the optical path from laser 21 and through the devices therealong described above, can, in the linear ramp portion of a serrodyne wave before its reset and well after its last reset, be represented as: ##EQU1## The parameter .theta. in this equation reflects any added phase shift due to the 90.degree. splice described above. The parameter .+-..phi..sub.r represents the Sagnac phase shift induced by rotation in one direction or another about the axis of symmetry of resonator 10 perpendicular to a plane passing through all of that resonator. The effective propagation constant in the foregoing equation, .beta., giving the effective phase change per unit length and the effective loss per unit length, comprises several terms, or .beta.=.beta..sub.o-1 -i.alpha.+.DELTA..beta.sin.omega..sub.m t. The term .beta..sub.o-1 =2.pi.n.sub.eff (f.sub.o +f.sub.1)/c, as indicated above, is the weighted average of the propagation constants of the two principal axes of birefringence of the optical fiber in resonator 10. This average is based on the fraction of travel over each axis by the electromagnetic waves in the resonator in the corresponding polarization state with axis changes due to the 90.degree. rotation splice in the optical fiber of that resonator as described above. A rotation of other than 90.degree. will give an uneven weighting to these axes. As was also described above, .alpha. is the coefficient giving the loss per unit length in the resonator optical fiber.
The parameter .DELTA..beta.=2.pi.n.sub.eff f.sub.m .DELTA..phi..sub.m /c is the change in the effective propagation constant due to the incoming electromagnetic waves having been modulated sinusoidally at the rate .omega..sub.m. Thus, although the last equation is indeed just for the counterclockwise traveling electromagnetic waves in resonator 10 reaching photodetector 22 that began in integrated optical waveguide 16, the counterpart equation for waves beginning in integrated waveguide 17 and traveling in the opposite direction in resonator 10 to reach photodetector 23 will be very similar, but will have the opposite sign for any rotation induced phase shift.
The foregoing equation for E.sub.d can be written more simply by making the substitution: ##EQU2## With the preceding substitution, the equation for the electric field component at photodetector 22 before a reset becomes: ##EQU3## Assuming that the completion of a reset in the serrodyne wave occurs at time t.sub.o, and that the transit time for the electromagnetic waves in resonator 10 from input coupler 11 to output coupler 12 over optical fiber length l.sub.1 is .tau..sub.1 with the transit time around the entire resonator being .tau., the electromagnetic wave reaching photodetector 22 in the time t.sub.o .ltoreq.t&lt;t.sub.o +.tau..sub.1 will be as follows: ##EQU4## that is, there is no effect at photodetector 22 during this initial transit time at photodetector 22. That is because the change in the input electric field component of the electromagnetic wave, which has now changed from E.sub.in to E'.sub.in, has not reached photodetector 22 yet in this time duration. The input electric field component of the counterclockwise electromagnetic wave after the reset of that wave can be written E'.sub.in =E.sub.in e.sup.i.phi..sbsp.s where .phi..sub.s represents the phase change due to the reset in the serrodyne part of the phase of the electric field component of the electromagnetic wave portion in integrated waveguide 16.
After an interval of .tau..sub.1, the change in the input electric field component of the counterclockwise electromagnetic wave reaches output coupler 12 for the first time but those portions of the input field component earlier introduced into resonator 10 prior to the reset continue to recirculate in resonator 10 with the phase they had at the introduction thereof, and they repeatedly reach output coupler 12. These earlier introduced portions still constitute the rest of the counterclockwise electromagnetic wave reaching input coupler 12 in the time duration immediately after t.sub.o +.tau..sub.1 following reset at t.sub.o. Until those portions dissipate, there will thus be a mixture of phases in the electric field component of the counterclockwise electromagnetic wave. Thus, the electric field component of the counterclockwise electromagnetic wave at photodetector 22 can be written after t.sub.o +.tau..sub.1 as follows: ##EQU5## or, following reset at t.sub.o, the representation for the electric field component of the counterclockwise electromagnetic wave will be: ##EQU6##
As can be seen in this last equation, there is clearly a delay before all of the electromagnetic wave portions traveling in one direction in resonator 10 fully represent a change in phase of the input electromagnetic wave giving rise to it. However, one can also see in the last equation that if the shift in phase in a serrodyne waveform at the end of a ramp is exactly 2.pi., or .phi..sub.s =2.pi., the equations after reset at t.sub.o essentially reduces to the equation before reset so that the equations describing events after reset are the equivalent of the equation describing events before the reset.
Assuming that .phi..sub.s =2.pi., so that the equations above describing the electric field component of the counterclockwise wave reaching photodetector 22 before and after reset are equivalent, the next few components in the feedback loop following photodetector 22 will provide an error signal representing deviations from resonance that is on the discriminant characteristic over frequency described above. Taking the equation above representing the electric field component at photodetector 22 prior to reset and establishing the definitions, ##EQU7## results in that equation being written: ##EQU8## The geometric progression in the summation on the right of the preceding equation can be put in the closed form known for such progressions to yield: ##EQU9##
The intensity of the counterclockwise electromagnetic wave impinging on photodetector 22 can be found, as is known from electromagnetic theory, by multiplying the electric field component thereof at photodetector 22 by its complex conjugate to give: ##EQU10## Thus, the output of photodetector 22 will be a current proportional to I.sub.d, and bias and amplification electronics 24 will then provide a voltage proportional to this current to be supplied at the input of filter 34.
Filter 34 can be a low pass filter having a cut-off frequency appropriately greater than .omega..sub.m, or it can be a bandpass filter which passes frequencies about .omega..sub.m. This is the frequency component to be selected from the output signal of bias and amplification electronics 24 as its amplitude will be a measure of the error, or the deviation from resonance of the electric field component of the counterclockwise electromagnetic wave in resonator 10.
The output of filter 34 is supplied to the signal input of phase detector 36. The output signal of bias modulation generator 28 is supplied to the demodulation input of phase detector 36, this signal being proportional to sin.omega..sub.m t. As a result, the output of phase detector 36 can be written: ##EQU11## where the output signal of the phase detector is the time average of (a) the signal at its signal input from the output of filter 34 represented as I.sub.d-filt and (b) the signal at its demodulation input, both multiplied together and having the average thereof taken over the period of the modulation signal, T.sub.m =2.pi./.omega..sub.m =1/f.sub.m. The constant G.sub.1 represents the effective gains of bias and amplification electronics 24, filter 34, and phase sensitive detector 36. This integral, as a function of f.sub.o +f.sub.1 occurring in the factor .beta..sub.o in .PSI., will provide an error signal, indicating by its value and algebraic sign where f.sub.o +f.sub.1 is with respect to resonance to thereby provide a discriminant characteristic over frequency for the corresponding feedback loop. This error signal directs that feedback loop to act to cause the counterclockwise wave in resonator 10 to take a frequency that permits it to be in resonance therein. Any errors along the discriminant function are stored in integrator 38 so that they may be corrected in this feedback loop. A similar result is reached for the feedback loop associated with the clockwise electromagnetic wave in resonator 10 provided to adjust the frequency f.sub.o +f.sub.2 so that wave also stays in resonance therein.
However, .phi..sub.s may well not equal 2.pi., even if initially set at that value, because of component aging, temperature variations, and the like. As a result, the equation for the electric field component of the counterclockwise electromagnetic wave reaching photodetector 22 after a reset therein at t.sub.o must be used in determining the system response in that situation, or ##EQU12## Making the same substitutions as were made above, that equation can be written as: ##EQU13## where the closed form for the geometric progressions involved have been used.
Once again, the intensity is found by multiplying the electric field component of the counterclockwise electromagnetic wave portion appearing in the last equation by its complex conjugate with the following result: ##EQU14##
As can be seen from the foregoing equation, the factor outside of the brackets is the same expression for the intensity reaching photodetector 22 found above in the situation of .phi..sub.s being equal to 2.pi.. Thus, the factor in the brackets represents the results of having .phi..sub.s .noteq.2.pi.. Because the factor outside of the brackets yielded a satisfactory discriminant characteristic over frequency, the factor within the brackets effectively represents an error in the discriminant characteristic which can lead to erroneous results of a degree depending on the deviation of .phi..sub.s from a value of 2.pi. and the number of recirculations which have occurred since the reset. Thus, there is a desire to have a resonant fiber optic gyroscope system assuring that the serrodyne waveform reset value is equal to 2.pi..